Structural Health Monitoring and Dynamic Identification of Structures: Applications

Transcription

1 Structural Health Monitoring and Dynamic Identification of Structures: Applications Computer Lab II, IMSS (017) Ground Work: Definition of the problem & System Modeling when we have full knowledge of the system characteristics and its initial conditions In the folder: System_ID, open the file System_Simulate.m Figure 1: (a) The two storey shear-frame and (b) The Kalamata Input Earthquake. Assume m1 = 100 Mgr m = 80Mgr and k1 = k = kn / m 1) Derive the mass and stiffness matrices for this degree-of-freedom (dof) system. Also, assume Rayleigh damping with the first two damping ratios being: ζ 1= ζ =5%. ) Calculate the damping matrix. This can he be derived as C=aMβΚ, where it holds that (from modal analysis theory): a βω ω ζ 1 = 1 1 βω ωζ a = ωζ ωζ β = 1 1 ω1 ω βω a = ω ζ (1) In the above expression are the first and second eigen-frequency of the structure. MATLAB s intrinsic function eig will be helpful for this implementation. 3) Bring the system into a state-space formulation. To do this, introduce a new vector, called the augmented state vector:

2 () Then use the governing equation of the system: to bring the system into this form (1 st order Ordinary Differential Equation form) (3) Also, introduce an observation equation, i.e., assume you would like to measure (observe) some component of the system response. Choose the 1st floor acceleration. (4) 4) Reproduce the response of the system assuming we KNOW that we start from 0 Initial Conditions and that we have PERFECT knowledge of the system. MATLAB s intrinsic function lsim will be helpful for this implementation. 5) Bring the system from continuous to discrete form: (5) Use MATLAB s cd command Assignment #: Parametric and non-parametric Identification methods A. The FDD method In the folder: System_ID, open the file RunFDD.m. 1. Define a white noise input (inp) using MATLAB s wgn or randn functions and resimulate the system response, maintaining the full vector of accelerations as output.. After appropriately setting the following input variables: in.fs=; %Sampling Frequency

3 in.nfft =; in.fc = ; in.npeaks =; %Number of Fourier Transform point used %Cutoff Frequency %Number of peaks you are looking for Run the FDD algorithm and identify the system s natural frequencies, out.fd, damping, out.z, and modal shape matrix, out.phi, and compare these to the theoretical (true) B. The ERA method In the folder: System_ID, open the file RunERA.m. 1. Define a white noise input (inp) using MATLAB s wgn function and re-simulate the system response, maintaining the full vector of accelerations as output.. After appropriately setting the following input variables: Nfft =; inptype=; % 'imp'=impulse, 'WN'=white noise, 'input'=known input order=150; %Define the size of the Hankel matrix - crucial for convergence Run the ERA algorithm and identify the system s natural frequencies, damping, and modal shapes and compare them to the theoretical ones. 3. Repeat the process above for the Kalamata Earthquake (known input) using the ERA C. The ARMA method In the folder: System_ID, open the file RunARMA.m. 1. Define a white noise input (inp) using MATLAB s wgn function and re-simulate the system response, maintaining the full vector of displacements as output.. After appropriately setting the following input variables: disp=; nmin=; nmax=; %Define the top floor displacement as measurement %Minimum order %Maximum order Run the ARMA algorithm and identify the system s natural frequencies, damping, and take a look at the delivered statistics on the prediction error and model order. What can you say on the model fit? 3. Repeat the process above for the Kalamata Earthquake (known input) using the ARMA model. What can you say on the model fit?

4 Assignment #3: The Kalman Filter & the Luenberger Observer The purpose of this exercise is to demonstrate how far we can go with predicting the response of the system under dynamic excitation under uncertainties. In the folder: luenbergerstateestimation_exercise, copy System_Simulate.m and open the file luenbergerstateestimationworkspace.m D. The Luenberger Observer How to track the system if we do not know the Initial Conditions? The Luenberger observer utilizes measurements from the system to provide an estimate of the state (displacement, velocity, acceleration) in real-time, i.e., immediately as measurements are obtained. 6) Open file luenbergerstateestimation.mdl - Click on the block named: system under observation Try to assemble the puzzle pieces so that the equation of the system is reproduced: (6) Keeping in mind that the notation of the blocks represents the following: Input & Output block Gain, i.e., multiplication with a scalar or matrix Addition block for two sources Unit delay: this takes the state (vector x) one time step back in time, i.e., from to - Click on the block named: system under observation

5 Try to assemble the puzzle pieces so that the Luenberger observer is reproduced: (7) 7) Once again go back to the file luenbergerstateestimationworkspace.m Explore the sensitivity of the observer to the choice of Initial Condition assumption, x0, and the construction of the gain (via changing eigvals) and observe the effect on simulation (press the green run button in the Simulink file) 8) Define a false system model by reducing the mass by 5%. Simulate the system again and notice the effect on the results. E. The Kalman Filter or how to track the system if we do not know the Initial Conditions, but additionally when the model of the system is uncertain (modeling error) In the folder: KFStateEstimation_exercise: 9) Open the file KF_simulation.mdl Notice the similarities and differences to the Luenberger observer. How is the probabilistic component now reflected? Why do we need to now speak in term of probabilities? In other words, where does the uncertainty come from? 10) Open the file KF_main.m - Locate the lines of code where the process noise matrix Q (4x4) is defined and assign some - Locate the lines of code where the process noise matrix S (x) is defined and assign some Run the Matlab script KF_main.m for different values of Q, S and explore the sensitivity of the filter performance to the choice of these 11) Define a false system model by reducing the mass by 5%. Simulate the system again and notice the effect on the results.